Smooth Fields of Hilbert Spaces, Hermitian bundles and Riemannian Direct Images
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913888451166208 |
|---|---|
| author | Belmonte, Fabian Bustos, Harold |
| author_facet | Belmonte, Fabian Bustos, Harold |
| contents | Given a field of Hilbert spaces there are two ways to endow it with a smooth structure: the standard and geometrical notion of Hilbert (or Hermitian) bundle and the analytical notion of smooth field of Hilbert spaces. We study the relationship between these concepts in a general framework. We apply our results in the following interesting example called Riemannian direct images: Let $M,N$ be Riemannian oriented manifolds, $ρ:M\to N$ be a submersion and $π:E\to M$ a finite dimensional vector bundle. Also, let $M_λ=ρ^{-1}(λ)$ and fix a suitable measure $μ_λ$ in $M_λ$. Does the field of Hilbert spaces $\mathcal{H}(λ)=L^2(M_λ,E)$ admits a smooth field of Hilbert space structure? or a Hilbert bundle structure? In order to provide conditions to guarantee a positive answer for these questions, we develop an interesting formula to derivate functions defined on $N$ as a integral over $M_λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04066 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth Fields of Hilbert Spaces, Hermitian bundles and Riemannian Direct Images Belmonte, Fabian Bustos, Harold Functional Analysis 46C99 Given a field of Hilbert spaces there are two ways to endow it with a smooth structure: the standard and geometrical notion of Hilbert (or Hermitian) bundle and the analytical notion of smooth field of Hilbert spaces. We study the relationship between these concepts in a general framework. We apply our results in the following interesting example called Riemannian direct images: Let $M,N$ be Riemannian oriented manifolds, $ρ:M\to N$ be a submersion and $π:E\to M$ a finite dimensional vector bundle. Also, let $M_λ=ρ^{-1}(λ)$ and fix a suitable measure $μ_λ$ in $M_λ$. Does the field of Hilbert spaces $\mathcal{H}(λ)=L^2(M_λ,E)$ admits a smooth field of Hilbert space structure? or a Hilbert bundle structure? In order to provide conditions to guarantee a positive answer for these questions, we develop an interesting formula to derivate functions defined on $N$ as a integral over $M_λ$. |
| title | Smooth Fields of Hilbert Spaces, Hermitian bundles and Riemannian Direct Images |
| topic | Functional Analysis 46C99 |
| url | https://arxiv.org/abs/2308.04066 |